Here's a rough mapping using induction principles. We are aiming to prove that $|S_{r+2}|-|S_r|\approx 6r^2+12r+8$. We take the area of the annulus, which is $\pi(r+2)^2-\pi r^2=\pi(4r+4)$. We claim that the number of lattice points inside this region is therefore about $\pi(4r+4)$, and using triangular numbers, the number of actual lines is approximately $\pi^2(4r+4)(4r+5)/2$, and we remove a factor of $\zeta(2)$ to allow for the coprimality condition, leaving $3(4r+4)(4r+5)=3(16r^2+36r+20)$, which is not a bad approximation to the $6r^2+12r+8$ we require.

The double ring structure comes from the introduction of a new ring. We get both (new ring $\to$ new ring) and (new ring $\to$ previous ring) entries. The rarifaction of the previous ring leads to the final smoothing of the polynomial, so all we now need to do is scale the factors involved.

I wrote some reasonable JavaScript to test this:

```
<!DOCTYPE html>
<html>
<body>
<span id='s'></span>
</body>
<script>
function dist(x,y) {
return Math.sqrt((x[0]-y[0])*(x[0]-y[0])+(x[1]-y[1])*(x[1]-y[1]));
}
function gcd(a,b) {
var x,y;
x=Math.abs(a[0]-b[0]);
y=Math.abs(a[1]-b[1]);
if ((x==0 && y>1) || (x>1 && y==0)) return 0;
var g,mx=1;
for (g=2;g<=Math.max(x,y)/2;g++) if (x%g==0 && y%g==0) mx=g;
return mx;
}
for (d=0;d<=25;d++) {
c=0;
for (i=-d;i<=d;i++)
for (j=-d;j<=d;j++)
for (a=-d;a<=d;a++)
for (b=-d;b<=d;b++) {
if (a<i) continue;
if (a==i && b<=j) continue;
if (dist([i,j],[0,0])<=d && dist([a,b],[0,0])<=d && (i!=a || j!=b) && Math.abs(dist([i,j],[0,0])-dist([a,b],[0,0]))<1 && gcd([i,j],[a,b])==1) c++;
//if (d==2) console.log([i,j]+'..'+[a,b]);}
}
console.log(c+'..'+Math.pow(2*d,3));
}
</script>
</html>
```